7-1 Introduction The field of statistical inference consists of those methods used to make decisions or to draw conclusions about a population. These methods utilize the information contained in a sample from the population in drawing conclusions. Statistical inference may be divided into two major areas: Parameter estimation Hypothesis testing

7-1 Introduction Definition

7-1 Introduction

7-1 Introduction

7-2 General Concepts of Point Estimation 7-2.1 Unbiased Estimators Definition

7-2 General Concepts of Point Estimation Example 7-1

7-2 General Concepts of Point Estimation Example 7-1 (continued)

7-2 General Concepts of Point Estimation 7-2.3 Variance of a Point Estimator Definition Figure 7-1 The sampling distributions of two unbiased estimators

7-2 General Concepts of Point Estimation 7-2.3 Variance of a Point Estimator Theorem 7-1

7-2 General Concepts of Point Estimation 7-2.4 Standard Error: Reporting a Point Estimate Definition

7-2 General Concepts of Point Estimation 7-2.4 Standard Error: Reporting a Point Estimate

7-2 General Concepts of Point Estimation Example 7-2

7-2 General Concepts of Point Estimation Example 7-2 (continued)

7-2 General Concepts of Point Estimation 7-2.6 Mean Square Error of an Estimator Definition

7-2 General Concepts of Point Estimation 7-2.6 Mean Square Error of an Estimator

7-2 General Concepts of Point Estimation 7-2.6 Mean Square Error of an Estimator Figure 7-2 A biased estimator that has smaller variance than the unbiased estimator

7-3 Methods of Point Estimation Definition Definition

7-3 Methods of Point Estimation Example 7-4

7-3 Methods of Point Estimation 7-3.2 Method of Maximum Likelihood Definition

7-3 Methods of Point Estimation Example 7-6

7-3 Methods of Point Estimation Example 7-6 (continued)

7-3 Methods of Point Estimation Figure 7-3 Log likelihood for the exponential distribution, using the failure time data. (a) Log likelihood with n = 8 (original data). (b) Log likelihood if n = 8, 20, and 40.

7-3 Methods of Point Estimation Example 7-9

7-3 Methods of Point Estimation Example 7-9 (continued)

7-3 Methods of Point Estimation Properties of the Maximum Likelihood Estimator

7-3 Methods of Point Estimation The Invariance Property

7-3 Methods of Point Estimation Example 7-10

7-3 Methods of Point Estimation Complications in Using Maximum Likelihood Estimation It is not always easy to maximize the likelihood function because the equation(s) obtained from dL()/d = 0 may be difficult to solve. It may not always be possible to use calculus methods directly to determine the maximum of L().

7-3 Methods of Point Estimation Example 7-11

7-3 Methods of Point Estimation Figure 7-4 The likelihood function for the uniform distribution in Example 7-11.

7-4 Sampling Distributions Statistical inference is concerned with making decisions about a population based on the information contained in a random sample from that population. Definition

7-5 Sampling Distributions of Means Theorem 7-2: The Central Limit Theorem

7-5 Sampling Distributions of Means Figure 7-6 Distributions of average scores from throwing dice. [Adapted with permission from Box, Hunter, and Hunter (1978).]

Example 7-13

7-5 Sampling Distributions of Means Figure 7-7 Probability for Example 7-13.

7-5 Sampling Distributions of Means Definition